Uniqueness of solutions for an elliptic equation modeling MEMS. (English) Zbl 1171.35044

Let \(\Omega\) be an open set in \({\mathbb R}^N\) and let \(f\) be a smooth nonnegative function defined on \(\Omega\). This paper is devoted to the study of solutions \(0<u<1\) of the nonlinear elliptic equation \(-\Delta u=\lambda f(x)/(1-u)^2\) in \(\Omega\), under the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The authors are mainly concerned with the role of the real parameter \(\lambda\), in relationship with the geometry of the domain \(\Omega\). There are established several existence and non-existence results and the proofs combine various tools, including the implicit function theorem and the Lyapunov-Schmidt reduction theorem.


35J60 Nonlinear elliptic equations
35B32 Bifurcations in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J20 Variational methods for second-order elliptic equations
35Q60 PDEs in connection with optics and electromagnetic theory
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